Having risen up the leagues to lead the line for his country, Lambert's heroics for the Three Lions have captured the nation's heart.
The 31-year-old has not yet managed to produce the same form for Saints, who are flying high in the Premier League after claiming three wins on the trot.
The latest came against Swansea on Sunday, with captain Lallana opening the scoring in a hard-fought, and somewhat fortuitous, 2-0 win at St Mary's.
It was the latest of several impressive performances from the Saints skipper, who manager Pochettino believes could well be next in line to represent England.
"I'm very happy for Adam," he said of Lallana, who was called up for England's World Cup qualifier with Ukraine in September 2012 but did not play.
"He's a great bloke and a great professional and he's very passionate about this club, about Southampton, so I'm happy for him.
"I'm very happy that he scored and for everything he represents - all the good values at this club.
"It's not my decision in the end, but if he keeps performing as he did today he'll be actually very close to getting that call up to the national side.
"I don't like to talk about what I've done in his game. It's an overall group of people who are helping to make this club better.
"I want to thank him for his kind words about me, but there's a structure at this club to help everyone improve and that goes from the chairman to the cleaning lady.
"The one person who deserves all the praise for his improvement is the player himself. His performance has been amazing over the last few games."